**ANALYSING AMINO ACIDS IN HUMAN GALANIN AND ITS **

**RECEPTORS - GRAPH THEORETICAL APPROACH **

**Suresh Singh G.a and Akhil C. K.b **

a

Department of Mathematics, University of Kerala, Kariavattom, Thiruvananthapuram – 695581, Kerala, India,

b

Department of Mathematics, University of Kerala, Kariavattom, Thiruvananthapuram – 695581, Kerala, India,

**Acknoldgements **

We wish to thank Dr Achuthsankar S. Nair, Head of the Department, Department of Computational Biology and Bioinformatics, University of Kerala, Thiruvananthapuram along with the faculty members, Vijayalakshmi B. and Biji C. L. for their fruitful suggestions and constant encouragements.

**Abstract: ** *Graph theoretical analysis is an important area of research in biological *
*networks. In this work we define some new graphs called bipartite Pt-graphs and *
*their physicochemical subgraphs for peptides/proteins and their receptors based *
*on the physicochemical properties of amino acids. Here we analyze bipartite *
*Pt-graphs and their physicochemical subPt-graphs of human galanin and its three *
*receptors graph theoretically. From the graph theoretical analysis of bipartite *
*Pt-graphs and the physicochemical subPt-graphs we get some observations about the *
*relations among the amino acids, physicochemical properties, galanin and its *
*receptors. By a graph theoretical parameter of physicochemical subgraphs we get *
*all the collections of maximum independent pairs of amino acids which connect *
*the galanin and receptors by sharing exactly * 𝑛 (𝑛 = 1,2,3, … ) * common *
*physicochemical properties. These analyses can be used to study all the *
*relationships between peptide/protein ligands and their receptors and this may *
*help in the field of drug designing. *

**1.** **Introduction **

Proteins are polymers of amino acids, with each amino acid residue joined to its neighbour by a specific type of covalent bond [3]. Twenty different types of amino acids are commonly found in peptide/protein. The sequence of amino acids in a protein is characteristic of that protein and is called its primary structure [3]. Peptides/proteins are the compounds of amino acids in which a carboxyl group of one is united with an amino group of another. Neuropeptides are peptides formed and released by neurons. They are involved in a wide range of brain functions.

Galanin is a neuropeptide of 30 amino acids in humans and 29 amino acids in other species [4]. It is expressed in a wide range of tissues including the brain, spinal cord and gut. Its signaling occurs through three G protein-coupled receptors. It is linked to a number of diseases including Alzheimer’s disease, epilepsy, depression, eating disorders, cancer, etc.

In [7], we can see so many graph theoretical applications in various fields. Amino acid network with in protein was studied by S. Kundu [5]. By using some physicochemical properties (Hydrophobicity, Hydrophilicity, Polarity, Non-polarity, Aliphaticity, Aromaticity and Charge (Positive and Negative)) of amino acids, the amino acid network was studied by Adil Akthar and Nisha Gohan graph theoretically [1]. The centralities in amino acid networks were used by Adil Akhtar and Tazid Ali [2]. By using the concept of amino acid network we defined and analysed the peptide/protein graph (Pt-graph) and species peptide/protein graph (SPt-graph) of galanin present in fourteen species of animals graph theoretically [6]. In this work we define and analyse new graphs - bipartite Pt-graphs and physicochemical subgraphs (physicochemical property-wise) - of human galanin and its three receptors on the basis of the physicochemical properties of amino acids. The maximum matching of physicochemical subgraphs is done to get all the collections of maximum independent pairs of amino acids which connect the galanin and its receptors by sharing exactly 𝑛 (𝑛 = 1,2,3, … ) common physicochemical properties of amino acids. This method can be applied for all relationships between peptides/proteins and their receptors and this may help in the field of drug designing.

**2.** **Basic Concepts of Graph Theory **

**Definition 2.1: **A Graph [7] 𝒢 is a pair 𝒢 = (𝒱, ℰ) consisting of a finite set 𝒱 and a set ℰ of
2-element subsets of 𝒱. The elements of 𝒱 are called vertices and elements of ℰ are called edges.
The set 𝒱 is known as the vertex set of 𝒢 and ℰ as the edge set of 𝒢. Two vertices 𝑢 and 𝑣 of 𝒢
are said to be adjacent, if an edge join 𝑢 and 𝑣, and two edges are adjacent if they have common
vertex. The number of vertices in a graph 𝒢 is called its order and the number of edges is its size.
A graph with 𝑝 vertices and 𝑞 edges is said to be a (𝑝, 𝑞) graph.

**Definition 2.2: **Centrality measures in Graphs [2] are the vertex representation which gives the
relative importance within the graph. A centrality is a real-valued function 𝑓 which assigns every
vertex 𝑣 ∈ 𝒱 of a given graph 𝒢 a value 𝑓(𝑣) ∈ ℝ.

**Definition 2.3: **Let 𝒢 be an arbitrary (𝑝, 𝑞) graph. ℳ ⊂ ℰ(𝒢) is said to be a matching in 𝒢 if its

be saturated by a matching ℳ (ℳ- saturated) or matched vertex with respect to ℳ if 𝑣 is incident with an edge of ℳ. Otherwise, the vertex is said to be unsaturated by ℳ (ℳ- unsaturated) or a single vertex with respect to ℳ. A matching ℳ of a graph 𝒢 is said to be a perfect matching if all the vertices of 𝒢 are saturated by ℳ.

**Definition 2.4:** A Pt-graph is defined as a graph 𝒢 = (𝒱, ℰ) of a peptide/protein in which the
vertex set, 𝒱 is the collection of all different amino acids presented in the peptide/protein and
weight of a vertex in 𝒢 is the number of times it appears in the sequence of the peptide/protein.
Two vertices are said to be adjacent in 𝒢 if they are consecutive elements in the sequence and
also have at least one common physicochemical property.

For all terminologies and notations not mentioned in this work, we follow [7] (related to graph theory) and [3] (related to biology).

**Remark: **Weight of a vertex implies the frequency of occurrence of a specific amino acid in a
sequence. Obviously greater the weight of a vertex of a Pt-graph implies greater the
characteristics of those particular amino acid can be attributed to the peptide/protein. Also the
centrality measures of a Pt-graph help us to determine the number of amino acids possess
inter-relationships with each other.

**3.** **Bipartite Pt-graphs and physicochemical subgraphs of human galanin and its receptors **

In this section we define some new graphs called bipartite Pt-graphs and physicochemical subgraphs for peptides/proteins and its receptors. Also we construct and analyze bipartite Pt-graphs and their physicochemical subPt-graphs of human galanin and its receptors using 𝒞-sets of corresponding Pt-graphs.

**efinition 3.1:** A bipartite Pt-graphis defined as a simple bipartite graph 𝒢 = (𝒱, ℰ) of a peptide/protein
and its receptors with 𝒳 and 𝒴 as the partitions of the vertex sets of the corresponding Pt-graphs of the
peptide/protein and its receptors respectively. Two vertices 𝑥 ∈ 𝒳 and 𝑦 ∈ 𝒴 are said to be adjacent if
they have atleast one common physicochemical property.

**Definition 3.2: **A 𝒞-set of a Pt-graph of a peptide/protein is defined as the subset of the vertex set
whose elements are the amino acids which recieve the highest centrality measures for each
physicochemical properties of amino acids.

**Figure 1: Pt-graph of human galanin **

**Figure 2: Pt-graph of receptor-1 **

**Figure 4: Pt-graph of receptor-3 **

From Pt-graphs we get the 𝒞-set from the highest centralities of amino acids for each
physicochemical property. For human galanin, 𝐺_{5} (Hydrophoic and Non-polar), 𝑆_{4} and 𝑁_{3}
(Hydrophilic and polar), 𝐿4 (Aliphatic), 𝑌1 and 𝑊1 (Aromatic), 𝐻2, 𝐾1 and 𝑅1 (Positive), 𝐷1

(Negative) are the amino acids which receive the highest centralities. Hence the 𝒞-set for the galanin is
{𝐺_{5}, 𝑆_{4}, 𝑁_{3}, 𝐿_{4}, 𝑌_{1}, 𝑊_{1}, 𝐻_{2}, 𝐾_{1}, 𝑅_{1}, 𝐷_{1}}. Similarly we get the 𝒞-sets for the Pt-graphs of receptor-1,
receptor-2 and receptor-3. Then the 𝒞-set for receptor-1 is {𝐴_{29}, 𝑁_{14}, 𝑆_{32}, 𝐿_{40}, 𝐹_{25}, 𝐾_{20}, 𝐸_{10}}, 𝒞-set
for receptor-2 is {𝐴_{46}, 𝑆_{28}, 𝑉_{30}, 𝐹_{17}, 𝑅_{27}, 𝐷_{8}, 𝐼_{18}, 𝑊_{8}} and 𝒞 -set for receptor-3 is
{𝐴_{65}, 𝐺_{32}, 𝑃_{25}, 𝑆_{19}, 𝐿_{50}, 𝐹_{8}, 𝑅_{39}, 𝐸_{8}, 𝐷_{9}, 𝑉_{28}}.

Next we analyse the bipartite Pt-graphs of human galanin and its receptors.

**Bipartite Pt-graphs of human galanin and its receptors **

**Figure 6: Bipartite Pt-graph of galanin and receptor-2 **

**Figure7: Bipartite Pt-graph of galanin and receptor-3 **

**Definition 3.3: ** Physicochemical subgraphs ℋ𝑘𝑖(for 𝑖 = 1,2,3, … ) of a bipartite Pt-graph 𝒢 of a
peptide/protein and 𝑘 receptors is defined as a subgraph whose vertex sets are same as that of 𝒢 and
two vertices in the different partitions are adjacent if they have exactly 𝑖 common physicochemical
properties.

**Remark: **Let 𝒢 = (𝒱, ℰ) be a bipartite Pt-graph of a peptide/protein and its 𝑘 receptors with 𝒳and 𝒴 as
the partitions of the vertex sets and let ℋ𝑘𝑖 𝒳, 𝒴𝑖 , where 𝒴𝑖 ⊆ 𝒴 (for 𝑖 = 1,2, … , 𝑛) be 𝑛
physicochemical subgraphs ,

Then,

𝒴𝑖 𝑖=1,2,3,…𝑛

= ∅ and 𝒴𝑖

𝑖=1,2,3,…𝑛

= 𝒴

**Physicochemical subgraphs of galanin and receptor-1 **

**Figure 8: **ℋ11**subgraph **

**Figure 9: **ℋ12**subgraph **

**Figure 10: **ℋ13**subgraph **

**Physicochemical subgraphs of galanin and receptor-2 **

**Figure 12: **ℋ21**subgraph **

**Figure 13:**ℋ22**subgraph **

**Figure 14: **ℋ23**subgraph **

**Physicochemical subgraphs of galanin and receptor-3 **

**Figure 16: **ℋ31**subgraph **

**Figure 17:**ℋ32**subgraph **

**Figure 18: **ℋ33**subgraph **

Next we obtain the maximum matching of ℋ𝑘𝑖 subgraphs of bipartite Pt-graphs of human galanin and its receptors. Table 1 represents all the collections of maximum independent pairs of amino acids which connecting the human galanin and its receptors by sharing exactly 𝑖 (𝑖 = 1,2,3,4) common physicochemical properties.

**Receptors **𝒌

**Maximum matching of **𝑯_{𝒌}𝒊** subgraphs **

𝓗𝒌𝟏** subgraphs ** 𝓗𝒌𝟐** subgraphs ** 𝓗𝒌𝟑** subgraphs ** 𝓗𝒌𝟒** subgraphs **

𝒌 = 𝟏

(𝐺_{5}, 𝐾_{20}), 𝑆_{4}, 𝐿_{40} ,
𝐿_{4}, 𝑆_{32} , 𝑌_{1}, 𝐸_{10} ,

𝐻2, 𝐹25 , (𝐾1, 𝐴29)

(𝐺_{5}, 𝑁_{14}), 𝑆_{4}, 𝐴_{29} ,
𝑌_{1}, 𝐿_{40} , 𝑊_{1}, 𝑆_{32} ,

(𝐻2, 𝐸10)

(𝐺_{5}, 𝐿_{40}), 𝐿_{4}, 𝐴_{29} ,
𝑊_{1}, 𝐹25 , 𝐻2, 𝐾20 ,

𝐾_{1}, 𝑁_{14} , 𝑅_{1}, 𝑆_{32} ,

𝑁_{3}, 𝐸_{10}

(𝐺_{5}, 𝐴_{29}), 𝑆_{4}, 𝑁_{14} ,
𝐿_{4}, 𝐿_{40} , 𝐾_{1}, 𝐾_{20} ,

𝐷_{1}, 𝐸_{10} , 𝑁_{3}, 𝑆_{32}

𝒌 = 𝟐

(𝐺_{5}, 𝐷_{8}), 𝑆_{4}, 𝑉_{30} ,
𝐿_{4}, 𝑆_{28} , 𝑊_{1}, 𝑅_{27} ,

𝐻_{2}, 𝐹_{17} , 𝐾_{1}, 𝑊_{8} ,

𝐷_{1}, 𝐴_{46} , (𝑁_{3}, 𝐼_{18})

𝑆_{4}, 𝐴_{46} , (𝐿4, 𝑊8),
𝑌_{1}, 𝐼_{18} , 𝑊_{1}, 𝑉_{30} ,

(𝐻_{2}, 𝑆_{28})

(𝐺5, 𝐼18), 𝐿4, 𝐴46 ,
𝑊_{1}, 𝐹_{17} , 𝐻_{2}, 𝑅_{27} ,

𝐾_{1}, 𝑆_{28} , 𝑅_{1}, 𝐷_{8}

(𝐺5, 𝐴46), 𝐿4, 𝐼18 ,
𝑌_{1}, 𝑊_{8} , 𝐾_{1}, 𝑅_{27} ,

𝐷_{1}, 𝐷_{8} , 𝑁_{3}, 𝑆_{28}

𝒌 = 𝟑

(𝐺5, 𝐷9), 𝑆4, 𝐿50 ,
𝐿_{4}, 𝑆_{19} , 𝑌_{1}, 𝑅_{39} ,

𝑊_{1}, 𝐸_{8} , 𝐻_{2}, 𝐹_{15} ,

𝐾_{1}, 𝐴65 , 𝑅1, 𝐺32 ,

𝐷_{1}, 𝑃_{25} , 𝑁_{3}, 𝑉_{28}

𝐺5, 𝑆19 , (𝑆4, 𝐴65),
𝑌_{1}, 𝑉_{28} , 𝑊_{1}, 𝐿_{50} ,

𝐻_{2}, 𝐸_{8} , (𝑁_{3}, 𝑃_{25})

(𝐺5, 𝐿50), 𝑆4, 𝑉28 ,
𝐿_{4}, 𝐴_{65} , 𝑊_{1}, 𝐹_{15} ,

𝐻2, 𝑅39 , 𝐾1, 𝑆19 ,

𝑅_{1}, 𝐸_{8} , 𝑁_{3}, 𝐷_{9}

(𝐺5, 𝐴65), 𝐿4, 𝑉28 ,
𝐾_{1}, 𝑅_{39} , 𝐷_{1}, 𝐸_{8} ,

𝑁_{3}, 𝑆19

**Table 1: Maximum matching of **𝓗_{𝒌}𝒊**subgraphs of human galanin and its receptors **

Let 𝒢_{𝑘} 𝒳, 𝒴_{𝑘} (for receptors 𝑘 = 1,2,3) be three bipartite Pt-graphs of human galanin and its
receptors, where 𝒳 is the vertex set of Pt-graph of human galanin and 𝒴_{1}, 𝒴_{2} and 𝒴_{3} are the vertex sets
of Pt-graphs of receptor-1, receptor-2 and receptor-3 respectively. Also let ℋ𝑘𝑖 𝒳, 𝒴𝑘 be the
physicochemical subgraphs of 𝒢_{𝑘} 𝒳, 𝒴_{𝑘} , where 𝑖 indicates the number of common physicochemical
properties of amino acids. Also let we denote the amino acids with physicochemical properties as

𝑃_{1}+= Hydrophobic, 𝑃1−= Hydrophilic,
𝑃_{2}+=Polar, 𝑃2−= Non-polar,

By analysing ℋ_{𝑘}𝑖 subgraphs, we get some observations about the physicochmical property-wise
connections of amino acids of galanin and its receptors.

**Observation 3.1: **In the analysis of the bipartite Pt-graphs of human galanin and receptors, we
obtain 𝑁_{14}, 𝑆_{32} (in receptor 1), 𝑆_{28}, 𝑊_{8} (in receptor 2) and 𝑆_{19} (in receptor 3) are the amino acids
which receiving all highest centralities. Also we obtain 𝐺_{5}, 𝑆_{4}, 𝑌_{1}, 𝑊_{1} and 𝑁_{3} are the common
amino acids of human galanin which receive the highest centralities in all the bipartite Pt-graphs.

**Observation 3.2: **In the physicochemical subgraphs with amino acids sharing exactly one
common property, the connections of amino acids of galanin to receptor-1, receptor-2 and
receptor-3 are

(1) 𝑃1+ is not connected with 𝑃1+ and 𝑃1− is not connected with 𝑃1− (2) 𝑃2− is not connected with 𝑃2−.

(3) 𝑃3+ is not connected with both 𝑃3+ and 𝑃3− but 𝑃3− is not connected with 𝑃3+. (4) 𝑃4+ and 𝑃4− are not connected with both 𝑃4+ and 𝑃4−.

**Observation 3.3: **In the physicochemical subgraphs with amino acids sharing exactly two
common properties, the connections of amino acids of galanin to receptor-1 and receptor-3 are

(1) 𝑃1+ and 𝑃1− are connected with both 𝑃1+ and 𝑃1− . (2) 𝑃2− is not connected with 𝑃2−.

(3) 𝑃3+ is not connected with, 𝑃3+, 𝑃3− and 𝑃30 𝑃3− is not connected with 𝑃3−

𝑃_{3}0 is not connected with 𝑃3+ and 𝑃3−.
(4) 𝑃4+ is not connected with 𝑃4−

𝑃_{4}− is not connected with 𝑃_{4}+, 𝑃_{4}− and 𝑃_{4}0
𝑃_{4}0 is not connected with 𝑃_{4}+ and 𝑃_{4}−.

**Observation** **3.4:** In the physicochemical subgraphs with amino acids sharing exactly two
common properties, the connections of amino acids of galanin to receptor-2 are

(1) 𝑃1+ and 𝑃1− are conected with both 𝑃1+ and 𝑃1−. (2) 𝑃2− is not connected with 𝑃2−.

(3) 𝑃3+ is not connected with 𝑃3+ and 𝑃30
𝑃_{3}− is not connected with 𝑃_{3}−
𝑃_{3}0 is not connected with 𝑃3+.
(4) 𝑃4+ is not connected with 𝑃4−

𝑃_{4}− is not connected with 𝑃4+, 𝑃4− and 𝑃40
𝑃_{4}0 is not connected with 𝑃_{4}+ and 𝑃_{4}−.

**Remark: **In the physicochemical subgraphs with amino acids sharing exactly two common
properties, the amino acids 𝑃3+ and 𝑃30 of galanin are conncted with the amino acids 𝑃3− of receptor-2

**Observation 3.5: **In the physicochemical subgraphs with amino acids sharing exactly three
common properties, the connections of amino acids of galanin to receptor-1, receptor-2 and
receptor-3 are

(1) 𝑃1+ is not connected with 𝑃1− and 𝑃1− is not connected with 𝑃1+. (2) 𝑃2− is not connected with 𝑃2+.

(3) 𝑃3+ and 𝑃3− are not connected with 𝑃3+. (4) 𝑃4+ is not connected with 𝑃4+.

**Observation 3.6: **In the physicochemical subgraphs with amino acids sharing exactly four
common properties, the connections of amino acids of galanin to receptor-1, receptor-2 and
receptor-3 are

(1) 𝑃1− have more neighbours than 𝑃1+ of 𝑋. (2) 𝑃2− have more neighbours than 𝑃2+ of 𝑋. (3) 𝑃4− have more neighbours than 𝑃4+ of 𝑋.

**Observation 3.7: **There is no neighbours for 𝑃_{3}− in receptor-2 of the physicochemical subgraph
with amino acids sharing exactly four common properties.

**Observation 3.8: **The physicochemical subgraph of galanin and receptor-3 with amino acids
sharing exactly one common property (figure 16) is the subgraph having a maximum matching
which is the only perfect matching among all the physicochemical subgraphs of galanin and its
receptors.

**Conclusion **

sharing exactly one common property (figure 16) is the subgraph having a maximum matching which which is the only perfect matching among all the physicochemical subgraphs of galanin and its receptors. These analyses can be used to study all the relationships between peptide/protein ligands and their receptors and this may help in the field of drug designing.

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